Interfering for the good of a chemical reaction

Abstract

Chemistry has, throughout its history, been a science of passive control: having made the conditions as favourable as possible, the chemist is obliged to let the complex reorganization of reactant molecules into products take place unhindered. But what if one could meddle at the very heart of a reaction to affect the dynamics of molecular evolution? Welcome to the world of quantum interference control.

Main

The use of chemistry to support human activity is very old. Think of an archaeological dig, and one immediately thinks of pottery fragments, nails, ancient coins, perhaps metal-tipped weapons or tools — ample evidence that exploitation of chemical operations preceded written history. In the earliest known written records, too, we find references to the preparation of fermented products such as wine and vinegar, and to dyeing and glass-making. It is clear that even in antiquity the amount of practical knowledge concerning chemical processes must have been considerable.

Of course, the relationships between different chemical processes were not yet recognized and organized, and the available knowledge could be transmitted only as practical rules of procedure. But the revolutionary idea that a particular substance can be converted into other substances — by heating, dissolving or mixing — must have spurred the search for new methods to control that transformation and steer it towards the desired products. The science of chemistry developed around that search, and its future will certainly include methods of reaction control unlike anything our forerunners could have imagined.

Over the past two centuries, chemistry has been set on a logical footing; intensive studies of synthetic methodology have given us many ways of generating a vast range of chemical species. The methods that are in current use rely on two fundamental procedures, or a combination of them. The first is the adjustment of a concentration ratio, either of reactants or intermediate species, so as to amplify the yield of the desired product. The second involves adjusting the rates of competing reactions that form different species from the same reactant (or intermediate) so as to enhance the formation of the one that is wanted. In practice, both methods are implemented by altering parameters such as the temperature, pressure, acidity or solvent composition. What these parameters have in common is that they are measures of the averaged properties of a system with many molecules; their use exemplifies passive control. I use the word 'passive' to convey the notion that the reactant molecules and any surrounding solvent molecules are not subjected to manipulation by external influences during the evolution from reactants to products.

Is such external control possible? We are all familiar with control of the behaviour of machines and other macroscopic systems based on the application of classical mechanics to engineering design. Less familiar are the examples of control of the microscopic behaviour of matter, for which quantum mechanics is the relevant description. However, only in the past few years has it become realistic to think about developing generic tools for active control of product formation in a chemical reaction. By 'active' I mean that, for example, external electromagnetic fields are used to alter the dynamics of molecular evolution during the reaction and thereby generate more or all of a particular product.

Controlling molecular motion

How is it possible to directly control the dynamics of evolution of a molecule? The property of matter that makes this possible is quantum interference, which arises from the fact that the motions of material particles, for example atoms and electrons, can be described in terms of the motion of waves¹. The tool that permits practical exploitation of quantum interference for control of molecular dynamics is the laser. Laser technology now permits the generation of very short pulses of light, shaped pulses of almost any type, pulses with a well-defined phase relationship, very pure monochromatic light fields, and very high intensity light fields.

To understand how quantum interference can help in active control requires one to abandon some mechanistic ideas associated with the classical description of a chemical reaction. For example, the notion that merely increasing the energy in a particular chemical bond will direct reaction to occur at that bond neglects two characteristic features of molecular behaviour. First, the nature of excitation by light absorption typically spreads the excitation over many bonds and, second, the rate at which energy is redistributed from an excited bond to other parts of the molecule is typically greater than the rate of reaction of that bond. In contrast, control of molecular dynamics through exploitation of quantum interference makes no assumption about the mechanism of energy flow in the molecule.

The quantum description of matter posits that there is a 'de Broglie wave' associated with every particle; it has a wavelength equal to Planck's constant divided by the particle momentum, and like any other wave it has an amplitude and phase. The particle's position cannot be pinpointed as in the classical description; only a probability distribution for the position can be specified. For example, in the lowest energy-state of the hydrogen atom, the electron is distributed spherically about the proton at a mean nonzero distance and with a wavelength equal to the equatorial circumference of the spherical distribution. The probability of finding a particle at any position is proportional to the square of the de Broglie wave amplitude at that position.

As a consequence of the wave character of matter, the possible energy states of electrons in a molecule are not distributed continuously; instead, they can assume only a discrete set of values, just as standing waves in a box can be only of certain frequencies.

These electronic energy states are associated with particular spatial distributions of the electrons of the molecule; electrons occupy a set of molecular orbitals, each of which has a particular spatial distribution and energy. In the lowest (ground) electronic state, N electrons fill the lowest energy N/2 orbitals, with each orbital having just two electrons.

Excited electronic states of a molecule are created when an electron is transferred from an occupied orbital to an unoccupied orbital, say by the absorption of light. Exciting an electron into a higher energy state corresponds to transferring de Broglie wave amplitude from one state to another. The probability of the transition is proportional to the square of the transition amplitude, that is, the integral of the product of the amplitudes for the two states, the electron displacement and the light field amplitude. It is this relationship between the probability of transition and the amplitude, coupled with the fact that molecular excitation or de-excitation may involve a succession of such transitions, that can generate quantum interference effects, as I shall explain.

There is a strong analogy between interference effects in a molecule and interference of light. Consider, for example, the 'two-slit' experiment familiar to countless schoolchildren, in which there is overlap of coherent waves (waves with a time-independent phase relationship) emanating from two point sources or slits. The effect of this overlap, viewed on a screen, is to produce brighter regions (wherever the amplitudes of the two waves have the same sign) and darker regions of decreased intensity (where the two waves cancel), thereby generating a spatial pattern of interference fringes. Just this kind of interference pattern can also be observed with electrons diffracted, for instance, by a crystal lattice. More important, there are analogous interference effects in a molecule. Whenever there are two or more independent excitation paths (sequences of transitions) between two states of a molecule, and these independent paths from state A to state B can be excited in a way that yields a constant relationship between their phases — that is, coherently — quantum interference will occur.

The overall probability of a transition from state A to state B is proportional to the square of the sum of the transition amplitudes for all paths that connect these states. Because there is also a phase associated with the amplitude along each excitation path, the differences between the phases of the several paths generate interference that affects the magnitude of the overall transition probability. Quantum mechanics tells us that it is meaningless to ask which paths are actually taken unless one of them is physically detected in a measurement — at which point the ability to exploit quantum interference between all possible paths is lost. But it is possible to influence the individual paths, and it is the realization of some form of control of the phases of these different excitation paths that permits control of the resulting overall transition probability and hence of the molecular dynamics.

Our visualization of atomic motion in a molecule is aided if we take advantage of the fact that the mass of an electron is about twenty-thousand times smaller than the mass of a typical atomic nucleus. Because of this disparity the electrons move much more rapidly than do the nuclei. The electrons adjust very quickly to any changes in the positions of the nuclei, so the latter in effect move in a field of electrical potential generated by the electrons, the energy varying with the relative positions of the nuclei. This potential field can therefore be mapped with respect to displacements of the nuclei (Fig. 1), and for typical molecules extends into many dimensions according to the number of nuclei; the resulting map is called the potential energy surface for that electronic state. In general, the potential energy surfaces of different electronic states are different.

Figure 1: Representation of the potential energy surface for the collinear reaction F+H₂ → H+HF.

The electronic energy of the three interacting atoms is plotted on the vertical axis. The axis pointing out of the page represents the distance between the H₂ molecule and the F atom; the axis pointing to the right represents the distance between the H atom and the HF molecule. The reaction can be envisaged as following a path along which the F atom approaches the H₂ molecule (into the page) until a closest approach arrangement FHH is reached, and then products HF and H separate along the axis pointing to the right. (Adapted from ref. 20.)

For many molecular systems the same global potential energy surface describes reactant and product molecules. The reactant molecule is identified with one of the minima of the potential energy surface. Elsewhere on the potential energy surface is another minimum, corresponding to the arrangement of atomic nuclei in the product molecule. A point on the potential energy surface can be used as a classical representation of the state of the molecule. The motion of that point on the potential energy surface can then be used to follow the changes in nuclear positions that accompany the transformation of a reactant into a product molecule. Sometimes the reaction must be followed on a different potential energy surface corresponding to an excited state.

It is not only the electronic motion in a molecule that is important. The small amplitude motions of the nuclei around a particular minimum of the potential energy surface generate a dense but discrete set of vibrational energy levels for that molecule. When fixed in position relative to one another, the overall motion of the nuclei generates a dense but discrete set of rotational energy levels for that molecule. A complete description of the molecule specifies its electronic, vibrational and rotational states.

We can exploit quantum interference to realize control of molecular dynamics in several superficially different ways. For clarity's sake, I will discuss only unimolecular reactions, that is, those situations in which a single reactant molecule is transformed to several product molecules.

Two-beam interference control

One way is to use two coherent monochromatic laser beams to create simultaneous excitations via different paths between two states of a molecule^2,3. This control method is sensitive to the structure of the potential energy surface only through the transition amplitudes between the states of the molecule. These transition amplitudes do depend on the nuclear positions.

Consider the case of a branching chemical reaction in which the excited reactant molecule can form at least two distinct product species. Suppose, for example, that two distinct excitation paths, in which the molecule is excited by absorbing, respectively, one and three photons, can connect the reactant and product states. Imagine that these excitation paths are activated by absorption of photons from two coherent light beams. Because the excitation sources are coherent, so too will be the amplitudes of the excitation transitions — an essential point, if the paths are to interfere. It is possible to modulate the interference between the coherent excitation amplitudes by changing their relative phase, and that is affected by changing the relative phase of their sources. The latter can be achieved by using, for the three- and one-photon excitations, the fundamental and the third harmonic wavelength beams of light derived from a single laser source. When these light beams pass through a gas (Fig. 2a), the dependence of refractive index on wavelength and the distance travelled in the gas can be used to tune the phase difference between them.

Figure 2: Two-beam interference control.

a, Experimental set-up (courtesy of R. J. Gurdon). b, Modulation of the DI⁺ and I⁺ signals as a function of phase difference between one- and three-photon pathways. The pressure of H₂ is a surrogate for the phase difference. (Adapted from ref. 21.)

Remember the classic two-slit experiment described earlier. In that case the amplitude of light reaching a screen was the sum of the amplitudes of waves from two different paths (the two slits). In the molecular example the excited-state wave amplitude is the sum of the excitation amplitudes generated by two routes that are not distinguished from each other by measurement. The result is, so to speak, an interference pattern in product space, with variations in phase difference giving larger or smaller chances of forming the desired product.

That, at least, is the theory. Execution of the experiments has proven to be very challenging, but was first demonstrated by Robert Gordon and his associates⁴ in studies of the branching reactions DI → D+I versus DI → DI⁺+electron. As shown in Fig. 2b, adjustment of the phase difference between the one- and three-photon excitation paths causes the concentrations of the products to oscillate. The maximum amount of D+I is generated (almost exactly) at the same phase as is the minimum amount of DI⁺, hence the ratio of concentrations of the products of the two reactions is controllable by that phase difference. A generalization of two-path interference control to multiple-path intense-field control is described in Box 1.

Two-pulse time delay control

A different realization of control of molecular dynamics can be achieved using a variable time delay between two extremely short light pulses⁵. A short light pulse can be represented as a coherent superposition of many monochromatic light waves; the range of the frequencies of the contributing light waves is inversely proportional to the duration of the light pulse. Because many of those frequencies can excite transitions, a very short light pulse applied to a molecule can simultaneously excite many coherent transitions to excited states. The state of the molecule thereby created consists of a coherent superposition of many excited states, corresponding to a superposition of de Broglie waves with many frequencies. A superposition of amplitudes of this type is called a wave-packet state, and its amplitude is usually localized in a small region of the excited-state potential energy surface of the molecule. The motion of the wave packet, and its control, does depend on the nature of the potential energy surface, and its variation with nuclear geometry. In other words, the way the wave-packet amplitude moves on the potential energy surface is sensitive to the presence of hills and valleys on the surface, respectively representing nuclear arrangements with higher and lower energies. At some time after the excitation pulse, because the frequencies of the components of the wave packet are different, the centre of the wave packet moves to a different position on the potential energy surface and its shape changes.

Consider the case that the excited-state wave packet moves to a new position and is then excited by a second pulse of light to a different potential energy surface. The time difference between the pulses can then be used to place amplitude into selected reaction channels and thereby control the branching to form different reaction products. The fashion by which interference affects this procedure can be seen as follows. Because each of the pulses contains many coherent frequency components, there are very many combinations of frequencies that lead from the initial state to the final state, each defining a path. The time delay between the pulses controls the relative phases of these combinations, and hence determines when the interference is constructive or destructive. Experimental studies reported by Gustav Gerber and associates⁶ show that in the branching reactions Na₂ → Na₂⁺+electron versus Na₂ → Na⁺+Na + electron, the ratios of concentrations of the products do indeed depend on the time delay between the pulses (Fig. 3).

Figure 3: Ratio of Na⁺ and Na₂⁺ ion signals as a function of pump–probe delay time.

(Adapted from ref. 22.)

Optimizing the control process

Two-beam interference and two-beam time delay are limiting cases of a more general approach to controlling molecular dynamics. Imagine continuously changing the frequency, amplitude and phase distributions throughout the duration of the applied pulse so as to continuously use interference between excitation paths to maximize the yield of a particular reaction product. The shaped pulse that accomplishes this goal is found by starting with the desired product state and the quantum mechanical equations of motion, and then working backwards in time to calculate the time distribution of electric field amplitude, frequency and phase that is required to maximize the product yield. A best (optimal) solution, subject to specific constraints (such as a restriction on the pulse energy or shape) can usually be found^7,8. The efficiency with which this pulse enhances the yield of a particular product is determined by the extent of interference between the amplitudes associated with its different spectral and temporal components.

Because the potential energy surfaces of molecules are complicated, the calculated optimal control pulse usually has a complicated distribution of frequencies, amplitudes and phases that change throughout its duration. Fortunately, this pulse optimization method lends itself to experimental automation⁹. Gerber and associates¹⁰, following up the seminal work of Herschel Rabitz and co-workers, have used a learning algorithm to optimize the phase distribution in a light pulse and thereby maximize the relative yield of one or the other of the products of two different bond-breaking reactions of the complicated molecule iron-cyclopentadiene-dicarbonyl chloride, CpFe(CO)₂Cl. This molecule is a metal–ligand complex of the kind that often acts as a catalyst for a biochemical reaction. For the purpose of this demonstration, the important property it has is that the different bonds, iron–cycopentadiene, iron–carbon monoxide and iron–chlorine, each have different strengths. The experimental results show that the pulsed light incident on a molecule can be optimized to maximize the yield of CpFeCOCl (obtained by cleavage of one of the FeCO bonds of CpFe(CO)₂Cl) relative to the yield of FeCl (obtained by cleavage of both FeCO bonds and the FeCp bond), or to minimize that yield.

Limits and applications

Active control of molecular dynamics for the purpose of optimizing product selectivity represents a paradigm shift in thinking about chemical synthesis. At first the suggestion that an optical field could control the evolution of a molecule by actively influencing the temporal development of the molecular dynamics, and thereby the choice of products in a chemical reaction, was greatly resisted. This resistance arose, in part, from the entrenched practice of describing reaction pathways using concepts from classical mechanics. It was augmented by the strongly held view that intramolecular energy redistribution is always so much faster than unimolecular fragmentation or isomerization that the ratio of products formed in a branching reaction is determined entirely by the statistical properties of the energy distribution in the reactant molecule. Furthermore, many researchers believed that generation of optical fields with the spectral and temporal characteristics required was not possible. It is the combined progress in introducing subtle ways of exploiting quantum interference, the explosive advances in laser technology and the successful execution of very complex experiments that has led to the current status of the field.

Despite its successful demonstration, it is unlikely that active control methods can replace any significant fraction of the well-honed synthetic procedures now in use. For one thing, most chemical reactions involve collisions between two molecules. Active control of the product yield in a bimolecular reaction requires control of the dynamics of the three-dimensional collisions of polyatomic reactants. Methods for achieving that control are in the earliest stage of development¹¹. In addition, most chemical syntheses are carried out in solution, and there remain theoretical and experimental uncertainties about the applicability of the currently available active control methods to reactions in solution. Perhaps most important, the cost of making precisely shaped laser pulses is high relative to the costs associated with current technology for carrying out reactions in solution.

Looking ahead

To what extent can we expect quantum interference control of molecular dynamics to find practical application?

I believe that laser technology will continue to advance and provide the tools necessary to implement theoretically sound active control methods. To a large extent then, the future for applications of active control of product will depend on the resolution of several fundamental theoretical issues. For instance, is there a limit to the attainable control of quantum dynamical processes? How does the efficiency of a control process depend on the number of degrees of freedom of the molecule? Is it possible to control bimolecular reactions? To what extent is the control of molecular dynamics possible when dissipation must be accounted for, as for a reaction in the liquid phase? How sensitive is the control field to fluctuations in the experimental environment and (from the point of view of prediction) to uncertainties in our knowledge of the molecular potential energy surface?

Partial answers to each of these questions are available, but better answers are needed. For example, the most powerful mathematical theorems available¹ concerning the limit to attainable control of molecular dynamics apply to the case when the possible states of the system are discrete and have energies that are all different. In that case, if the path from the initial to the final state is unrestricted, meaning that any sequence of states can be used to connect the initial and final states, it is possible in principle to transfer 100% of the molecules to any selected final state. Because most chemical reactions involve molecules that have both states with energies separated from one another by discrete amounts and states distributed continuously with respect to their energy, these theorems provide little guidance as to the limit of controllability of product selection.

Nor does demonstration that a light pulse can be tailored to optimize the yield of one product in a branching reaction establish the limit to the efficiency obtainable. Introducing adaptive learning in the active control process is one means of compensating for uncertainties in our knowledge.

The size of a molecule is relevant to the design of the process that controls its molecular dynamics for the following reason. As the number of atoms in a molecule increases so does both the number of energy levels per unit of energy and the complexity of the atomic motions associated with those energy levels. For active control of the dynamics of a polyatomic molecule to succeed, the coherence of the subset of excited states prepared by the laser pulse must not be rapidly destroyed by interactions with other levels of the molecule. The most recent experimental evidence (for which there is not yet a satisfactory theoretical description) strongly suggests that in many polyatomic molecules this loss of coherence can be relatively slow on the timescale of the control laser pulse¹², increasing the chances of success.

What about active control of the product yield in a bimolecular reaction? Control of the dynamics of a bimolecular collision can in principle be achieved by arranging for interference between different pathways connecting the same initial and final states of the colliding molecules. This task is very demanding, but there are indications from theory¹¹ that some special choices of the centre-of-mass motions of the reactants will generate the needed interference.

Reactant molecules in solution are constantly battered by solvent molecules, and these collisions ultimately destroy the coherence of the excited states necessary to actively control product formation. The key to active control of reactions in solution is reducing the influence of collisions between reactant and solvent molecules on the timescale needed for establishment of the interference that generates selectivity of product formation. The little available information concerning the timescale on which reactant–solvent collisions destroy excitation coherence suggests that in many cases it is several picoseconds. Because the lengths of optimized control pulses are around half a picosecond, there is a window for successful competition between retaining the coherence necessary for active control and destruction of that coherence.

Because the underlying principles of quantum interference control of molecular dynamics are broadly applicable, it is likely that applications in a variety of other fields will be developed in the near future, probably before practical applications in chemistry. One field in which this is already starting to happen is optoelectronics. For example, it is possible to use laser light to control the state of a single quantum dot (a nanometre-sized sample of metal or semiconductor with widely spaced energy levels), and interference effects can thereby be used to generate a fast semiconductor optical switch^13,14,15. It has also been shown that shaped light pulses can greatly enhance the yield of soft X-rays from gaseous argon and to alter in a programmed way the frequency distribution of the X-rays¹⁶.

Commercial prospects

Although the cost of producing laser light is very high, that may not rule out the use of active control methods for commercial production of chemicals. At present, photochemistry initiated by incoherent light is used industrially in only a few cases for the production of high-value intermediate and end-product chemicals, usually pharmaceuticals or perfumes. Because the cost of producing laser light is dropping, the possible advantages that accrue from the cleanliness of enhanced product generation by controlled laser pulses, for example by reduction in the need for expensive separation processes, may yet change the economic balance for some commercial products.

Going further into the realms of speculation, one can imagine developing active control methods for generating selective reactions of high-value added molecules, and for the design of new (not naturally occurring) biologically active molecules.

Peering still further into the future, if we can achieve active control of the selective reaction of biologically active molecules in solution, one can imagine developing a medical procedure that, using a fibre optic probe in situ, destroys by controlled reaction a specific molecule essential for the progression of a disease. Pipe dreams, perhaps; but when one sees how far we have come since the mix-it-heat-it-and-see days of early chemistry, maybe not so impossible a project after all.

Box 1 Multiple-path intense-field interference control

One of the most promising and most efficient methods for controlling molecular dynamics is based on the observation that interference between multiple paths connecting the same initial and final states applies to all kinds of transitions^17,18,19. Suppose, for example, that the relevant reactant and product states are as shown in the figure opposite. States 3 and 4 represent different products generated from the reactant, and these states have the same energy. One of the tasks of the control method is to selectively generate the product corresponding to state 3 rather than that corresponding to state 4, or vice versa. Imagine exciting the reactant molecule with a pulse of light from state 1 to state 2, and then with another pulse of light from state 2 to the product states 3 and 4. The product branching ratio will be determined by the ratio of the intensities of the transitions 2→3 and 2→4.

But now suppose the molecule is also irradiated with a third long-duration, strong pulse that has the correct wavelength to cause transitions between states 3 and 4 and a further reactant state 5. This pulse creates two coherent superpositions of states: 3 and 5, and 4 and 5. So now we have an indirect pathway to reaction, one that takes advantage of the cycling of amplitude in the coherent superposition of states 3 and 5 and of that in the coherent superposition of states 4 and 5. The frequency of that cycling increases as the pulse strength increases. The amplitudes associated with each of the multiply cycled paths, for example 1→2→3→5→3→5→…→3 and 1→2→4→5→4→5→…→4, can interfere with the amplitudes associated with the direct paths to products 3 and 4 respectively.

The pulse sequence required is counterintuitive. The pulsed field promoting the transitions 3→5 and 4→5 must be on while the pulsed fields that promote the transitions 1→2, 2→3 and 1→2, 2→4 are applied. Also, the pulsed field that promotes the transitions 2→3 and 2→4 must be applied before (but overlap with) the pulsed field that promotes the transition 1→2 so that when the coherent superposition states are subsequently coupled to state 1, new states are formed from which the pulse cannot transfer population to the intermediate state 2. Rather, the population is channelled directly into the product states. The relative populations of the product states are inversely related to the transition strengths between states 3 and 5 and states 4 and 5, so that by making different choices for state 5 one can control the ratio between the products.